MARKING POINTS FOR CRANK·ROCKER LINKAGE ON THE CENTERPOINT CURVE
By
E.
FILEMONDepartment of Applied Mechanics, Technical University, Budapest (Received November 26, 1970)
Presented by Prof. Dr. Gy. BEDA
The theoretically possible four bar linkages achieving the prescribed four positions of the coupler plane can be determined by means of the center- point and circlepoint curves. Using these Burmester curves by synthesis, it appears that some of the solutions fulfil the prescribed geometrical conditions theoretically but not constructionally. Namely, these mechanisms - although mountable in the four prescribed positions - either cannot move continuously between the four prescribed positions or the movement is continuous but the order of the positions is 'Hong. After the first unsuccessful trial, new frames have to be chosen until the constructionally correct solution [1]. But in some cases there is no solution to fulfil the conditions and therefore all attempts could get frustrated.
A survey of possible solutions could help to select two points on the centerpoint curve which directly result in a working mechanism. The proposed new diagram has not yet been published in the literature. This diagram be- longs to the four prescribed positions of the plane and together ,dth the center- point and eirclepoint curves, it characterizes all the solutions without marking a definite rotation center.
This method may be applied before computation and by limiting the number of possibilities, it may considerably reduce the running time. It is useful either as self-contained graphic procedure or can be built into the com- puter program.
Fig. la shows the four prescrihed positions of the coupler plane C and the resulting Burmester curves. In this present case, thc curves have single branches, which tend to infinity, but the method lends itself for two discon- nected branches, one closed and the other tending to infinity. To graphically determine or calculate the centerpoint curve is a tedious procedure
[2-6].
For given opposite pole quadrilateral it is possihle and helpful to prepare a sketch of the centerpoint curve to relieve the uncertainty of the order of connecting the availahle points and to indicate how to change the hasic data as necessary.[7 ---9]. To every revolute centerpoint on the centerpoint curve belongs a link R and the angles rjj12rjj13(J\.1 (Fig. Ib). 'Wether a link belonging to an arhitrarily
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JfARKL\-C FOISTS 289
chosen revolute ccntel', will be a crank or a rocker can only be dctermined after having choosen the second revolute cent er of the four bar linkage. But it is immediately evident - without marking any revolute center - what are the parts of the centerpoint curve with points unsuitable for a crank because the order of the positions would be wrang.
Two arbitrary revolute ccnters will determine all dimensions of the desir- ed linkage (fig. le). Thc Grashof condition can be checked with dimensions RI, R2, C and D.
The movement of plane C, starting from one of the possible initial posi- tions, is a geometrically periodical movement. All the possible equivalent ini- tial positions which produce the same movement of the plane C are all the positions of one possible periodical movement of the plane C. In the Burmester synthesis problem it is primordial that all the arbitrarily chosen four prescrib- ed position should be equivalent initial positions. All the initial positions are equivalent if the Grashof conditions, relating to the existence of a crank in the four bar linkage, is not satisfied. In these cases the linkage is a double rocker.
If this type of solution is suitable the problem is much simplified. Else there are t"<;\70 different periodical movements and the linkage can be either a double rocker, a double crank or crank and rocker type. In many cases a crank and rocker type solution is desired and as not all possible initial positions are equi- valent, further checking is needed.
The problem is: how to find a crank and rocker 'which will 'work well in practice?
Three conditions need to be fulfilled:
- The crank must go over the foul' prescibed positions in the right order:
All the four prescribed positions must be equivalent initial positions.
This means that all the positions of the rocker must be either above the frame or below it (Fig. le);
- The Grashof condition is satisfied and the crank must be the shortest link.
Can the first revolute center designated so that the pertaining link is a crank?
Fig. 2 is a graph of (]j12' (]jJ3' (]ju and R as a function of the arch length of the centerpoint curve. It appears that that part of the centerpoint curve will be suitable as a revolute center for operational cranks where either (]j12
<
<
(]j13<
(]j14 (anti-clockwise rotation) or(]j12> <
CP13> <
(]j14 (clockwise rota- tion). Functions appear to cross each other only at intersectionsQ
of the oppo- site sides of the opposite pole quadrilaterals. PointsQ
divide the centerpoint curve into six parts. It is clear vvithout determining the functions (]j, that if there is one point between two pointsQ
where the order (]j is good, all points of this section will be good, and vice versa. Two bad sections can be next to each other, but two good ones never.2!JO
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-'lARKING POINTS 291
There will be no rocker of crank and rocker linkage along that section of the centerpoint curve, where the graph (jj is partly below and partly above the line of 180:). If all the graphs are above the line of 1800 (and 1>12 < (jj13 <Wll )
the rocker will turn anti-clock:wise and if all the graphs Ware below the line of 1800 (and W12
>
< 1>13>«
11 ) the rocker will turn clockwise, between positions 2, 3 and "1.Any point on the marked section ,\ill he the intersection point of the frame directions for an infinite numher of solutions. Taking this point as a base, all the remaining points of the centerpoint curve define an angle Wo' Determining the graphs Wo and Wo 1800 the tcrritory between them must not he crossed hy graphs (jjw W13 and W1 .J if the four prescribed positions of the plane C are to be cquivalent initial positions.
It is possible that from the included zone, it appears that there is no rocker t ) operate satisfactorily with the marked crank.
A new· rocker could only be marked after choosing a new revolute center and plotting the new graphs Wo and Wo
+
1800, hencc hy trial and error meth- od.It
is easier to check a crank hclonging to a given rocker. Hence it is ad- visahle to choose the rocker first. The shaded area in :Fig. Id helonging to any arhitrary revolute cent er of rocker is inconvenient as a revolute center of a crank. The shaded area can he determined from the two extreme rocker posi- tions, belonging to the four prescribed positions. Using an angle meter the whole centerpoint curve can he surveyed for marking points of the centerpoint curve suitahle as revolute centers of a rocker.As a final test, only these parts need he examined to see if they satisfy the Grashof condition. In the example, the Grashof condition is not satisfied, there is not a single crank and rocker solution, 'which ,vould he feasihle.
In other cases, after the final test, there might he a finite section of the centerpoint curve the points of which contain good solutions. One among them can he chosen arhitrarily or hy the hest possihle ratio of WIZ' W13 and
1>14 of the crank, or in accordance with a suitahle optimum criterion.
The Grashof condition cannot he checked first to every arhitrary revo- lute center hecause different graphs of Grashof condition are inhihitively lahour- some and inaccurate to he assumed again and again for every point of the cen- terpoint curve.
The whole process can he computerized. Computer plotting of Rand angles W12' W13 and (P14 simplifies and increases the accuracy of the additional control.
Two revolute centers for other than crank and rocker type four har linkages can he marked similarly.
3 Periodica Polytcchnica ~L XV/3.
E. FILE.11OS
Summary
Formerly the only solution proposed for identifying frame points, for crank-rocker linkages, was to choose arbitrarily and to proceed by the trial and error, This paper suggests the use of a new diagram demonstrating all the possibilities simultaneously, offering a solu- tion immediately or after a far fewer trials, On the other hand. the diagram shows if there is no suitable linkage, making all trials useless.
References
1. HARTE),;BERG. R. S.: Denavit. 1.: Kinematic synthesis of linkages. :31cGraw-Hill, London, 1964.
2. PRI:lIOSE, E. J. F.: Frendstein, F., Sandor, G. ~.: Finite Burmcster theory in plane kine- matics. Journal of Applied :3Ieehanics, 633 (1964-).
3. TCHEHK1:DI)';OY, S. A.: Sintes ploskih sharnirnorichashnic mekhanisl11ov. Isdatclstvo Aka- demi Nauk. S. S. S. R. Moscow 1959.
,1. BEYER. R.: Kincl11atischc Getriebcsynthese. Springer-Verlag. Bcrlin 1953.
5. S"\.LY!' 1. J1:),;.: Some connections of the special cases of the Burmester designing process.
(In Hungarian). ~iiskolci Xchezipari }Hiszaki Egyetem Kozlel11enyci 12 (1963). pp.
·t33-452.
6. TERPL"\!'<. Z.: Mechanisms. (In Hungarian.) Tallkony\'l;:iad6, Budapest 1959.
7. FILE:lIO)';, E.: Conformal sketching of the centerpoint curve. (In Hungarian.) }1{iszaki Tudomany 42, 305-321 (1970).
8. FILE:\IO)';. E.: How to draw a tangent to any point of the centerpoint curve. Periodica Poly- technica }1 13, 1- 3 (1969).
9. FILE:lION, E.: Some questions of four-bar coupler-point curves. (In Hungarian) IUTA~I
Confercnce N agymaros 1968.
Dr. Elisaheth FILE::\ION Budapest
